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Liquid convection effects on the pushing-engulfment transition of insoluble particles by a solidifying interface: Part II. Numerical calculation of drag and lift forces on a particle in parabolic shear flow

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In this work, lift and drag forces acting on a particle in the close vicinity of a wall are calculated by numerically solving the incompressible two-dimensional Navier-Stokes equations. The flow field computations are done using the well-known Marker and Cell (MAC) method on a staggered grid. A parabolic shear flow at the inlet is assumed. The particle is assumed to be nonrotating and Magnus forces are not considered. The numerical results are compared to those obtained from analytical/empirical expressions for drag and lift forces from different theoretical models. Reasonably good agreement has been found between the two approaches.

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A P :

surface area of the spherical particle

C L :

lift coefficient

C D :

drag coefficient

D :


F D :

drag force

F S :

Saffman force

F xp , F yp :

force due to pressure effect


force due to viscous effect

R I :

radius of curvature of the interface

R p :

radius of the particle


flow Reynolds number

Re p :

particle Reynolds number

V o :

maximum convection velocity

V rel :

velocity of the particle relative to the liquid

V Lx :

liquid velocity in the x direction

V P :

particle velocity

d :

distance between the particle and the solid/liquid interface

i,j :

location of grid points

k a :

function of thermal conductivities

P :


t :


u, v :

fluid velocity along the x and y directions respectively

Δx, Δy :

grid sizes in x and y directions (Cartesian coordinate system)

Δr, Δz :

grid sizes in r and z directions (cylindrical coordinate system)

Δt :


α :

interface curvature in BC model

δ :

boundary layer width

η :

dynamic viscosity of the melt

v :

kinematic viscosity of the melt

ρ :


τ :

shear stress

Δρ :

density difference

I :


L :


P :

particle/phase field

S :



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Mukherjee, S., Sharif, M.A.R. & Stefanescu, D.M. Liquid convection effects on the pushing-engulfment transition of insoluble particles by a solidifying interface: Part II. Numerical calculation of drag and lift forces on a particle in parabolic shear flow. Metall Mater Trans A 35, 623–629 (2004).

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